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topology (now called analysis situs) as first developed bore but
little resemblance to the elaborate theory which to-day absorbs
all the energies of a prolific schools it may be of interest to state
the trivial puzzle which apparently started the whole vast and
intricate theory. In Euler s time seven bridges crossed the river
Pregel in Konigsberg, as in the diagram, the shaded bars repre-



senting the bridges. Euler proposed the problem of crossing all
seven bridges without passing twice over any one. The problem
is impossible.
The nature of Riemann's use of topological methods in the
theory of functions may be disposed of here, although an ade-
quate description is out of the question in untechnical language.
For the meaning of "uniformity' with respect to a function of a
complex variable we must refer to what was said in the chapter
on Gauss. Now, in the theory of Abelian functions, multiform
functions present themselves inevitably; an n-valued function
of s is a function which, except for certain values of z, takes
precisely n distinct values for each value assigned to s. Illus-
trating multiformity, or many-valuedness, for functions of a real
variable, we note that y, considered as a function of a?, defined
by the equation y* = x, is two-valued. Thus, if y  4, we get
t/2 = 4, and hence ^ = 2or2;ifaiis any real number except
zero or 'infinity', y has the two distinct values of Vr and  V#.
In this simplest possible example y and x are connected by an
algebraic equation, namely yz ~ x = 0. Passing at once to the
general situation of which this is a very special case, we might